RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

368 Chapter 19. The Paris memoir
2. If all the roots y1, . . . , yn have integer degrees,
and
the formula became (ε = n)
γ = 1 +
= 1 +
�
n τ−1
∑
τ=1
n
∑
hy1, . . . , hyn ∈ Z,
n1 = · · · = nε = 1,
∑ mν +
ν=1
mτ − 1
2
τ−1
∑
τ=1 ν=1
�
mν − n = 1 − n +
−
n
mτ + 1
∑ 2 τ=1
n
∑ (n − τ) mτ.
τ=1
It is interesting to consider the usefulness of these examples. It appears that the
examples were both chosen because the assumptions made therein corresponded to
particularly interesting cases and because they illustrate cases, in which the rather
complicated formula (19.28) — which looked even more complicated in ABEL’S nota-
tion than in my modern one — reduced to extremely simple forms. The first class of
equations considered (in which all degrees were equal) contains equations such as
χ (x, y) = y n − p (x) = 0
in which p is a polynomial. The second assumption (all roots have integer degrees)
applies to equations of the form
χ (x, y) = ∏ k
(y − p k (x)) = 0.
The indeterminates a, a ′ , a ′′ , . . . . ABEL chose to designate by α the number of inde-
terminates a, a ′ , a ′′ , . . . and ventured to investigate the relationships between the roots
x1, . . . , xµ and the indeterminates a1, . . . , aα. To the α indeterminates corresponded α
equations
θ (yτ) = 0 for τ = 1, . . . , α
which were linear in the indeterminates (see box 19.2.3, above). These equations “in
general” served to express the indeterminates rationally in x1, . . . , xα and y1, . . . , yα.
Only in cases of multiple roots would the equations not suffice. In such cases ABEL
involved the calculus which could be used to produce a set of α independent equations
for determining a1, . . . , aα. When ABEL divided F (x) by ∏ α τ=1 (x − xτ), he obtained
another equation
F1 (x) =
∏ α τ=1
F (x)
= 0
(x − xτ)